2006/11/16 by Oliver Díaz-Espinosa, Oliver Diaz-Espinosa, Diaz-Espinosa, Oliver +2
Computer Science · Mathematics · Physics and Astronomy · #37C30 #37E20 #60F05 #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Nonlinear Dynamics and Pattern Formation #Quantum chaos and dynamical systems #math-ph #math.DS #math.MP #msc:37C30 #msc:37E20 #msc:60F05
paper · pdf · doi:10.48550/arxiv.math/0611514
79 pages
arxiv created 2006/11/16 · openalex publication_date 2006/11/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study of the effect of weak noise on critical one dimensional maps; that is, maps with a renormalization theory. We establish a one dimensional central limit theorem for weak noises and obtain Berry--Esseen estimates for the rate of this convergence. We analyze in detail maps at the accumulation of period doubling and critical circle maps with golden mean rotation number. Using renormalization group methods, we derive scaling relations for several features of the effective noise after long times. We use these scaling relations to show that the central limit theorem for weak noise holds in both examples. We note that, for the results presented here, it is essential that the maps have parabolic behavior. They are false for hyperbolic orbits.